Simplify and expand the following expression: $ \dfrac{4}{q + 4}- \dfrac{5}{2q - 10}- \dfrac{5}{q^2 - q - 20} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{5}{2q - 10} = \dfrac{5}{2(q - 5)}$ We can factor the quadratic in the third term: $ \dfrac{5}{q^2 - q - 20} = \dfrac{5}{(q + 4)(q - 5)}$ Now we have: $ \dfrac{4}{q + 4}- \dfrac{5}{2(q - 5)}- \dfrac{5}{(q + 4)(q - 5)} $ The least common multiple of the denominators is: $ (q + 4)(q - 5)$ In order to get the first term over $(q + 4)(q - 5)$ , multiply by $\dfrac{2(q - 5)}{2(q - 5)}$ $ \dfrac{4}{q + 4} \times \dfrac{2(q - 5)}{2(q - 5)} = \dfrac{8(q - 5)}{(q + 4)(q - 5)} $ In order to get the second term over $(q + 4)(q - 5)$ , multiply by $\dfrac{q + 4}{q + 4}$ $ \dfrac{5}{2(q - 5)} \times \dfrac{q + 4}{q + 4} = \dfrac{5(q + 4)}{(q + 4)(q - 5)} $ In order to get the third term over $(q + 4)(q - 5)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{5}{(q + 4)(q - 5)} \times \dfrac{2}{2} = \dfrac{10}{(q + 4)(q - 5)} $ Now we have: $ \dfrac{8(q - 5)}{(q + 4)(q - 5)} - \dfrac{5(q + 4)}{(q + 4)(q - 5)} - \dfrac{10}{(q + 4)(q - 5)} $ $ = \dfrac{ 8(q - 5) - 5(q + 4) - 10} {(q + 4)(q - 5)} $ Expand: $ = \dfrac{8q - 40 - 5q - 20 - 10}{2q^2 - 2q - 40} $ $ = \dfrac{3q - 70}{2q^2 - 2q - 40}$